A MAXIMUM ENTROPY APPROACH FOR UNCERTAINTY QUANTIFICATION OF INITIAL GEOMETRIC IMPERFECTIONS OF THIN-WALLED CYLINDRICAL SHELLS WITH LIMITED DATA
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摘要: 具有不确定性特征的初始缺陷被认为是导致薄壳结构实际临界载荷值与理论解不相符并呈现分散特征的主要原因. 对实际薄壳结构初始缺陷的建模至少需要考虑两个方面的不确定性量化, 一是对缺陷分布形式和幅值等固有随机性的量化, 二是对小样本量和不准确测量所导致缺陷统计量的不确定性的量化. 本文在利用随机场的Karhunen-Loeve展开法对薄壳初始几何缺陷建模的基础上, 提出一种基于极大熵原理的缺陷建模方法. 首先, 采用极大熵分布来估计Karhunen-Loeve随机变量的概率密度函数, 以适应不能使用高斯随机场进行缺陷随机场建模的情况. 随后, 通过将经典的等式约束极大熵模型扩展为区间约束极大熵模型, 实现对实际工程中仅能获得少量薄壳结构几何缺陷样本数据所导致的认知不确定性的量化. 最后, 将所提方法用于对国际缺陷数据库的A-Shell进行缺陷建模和临界载荷预测. 研究表明, 所提基于区间约束极大熵原理的随机场建模方法在能够有效表征实测数据高阶矩信息的同时, 还具备量化小样本数据导致的认知不确定性的能力, 并且高斯随机场模型和基于等式约束极大熵原理的随机场模型是本文所提建模方法的两种特殊情况.Abstract: Initial imperfections with uncertainty characteristics are well-known recongnized as the main reason why the actual critical load values of thin-shell structures do not match the theoretical solutions and exhibit dispersion characteristics. In order to model the initial imperfections of actual thin-shell structures more appropriately, seveal sources of uncertainty quantification should be handled carefully, such as the quantification of the inherent randomness in the form and magnitude of the imperfection distribution, and the quantification of the uncertainty in the statistics due to small sample size and inaccurate measurements in practice. In this paper, a novel modeling approach for initial geometric imperfections of thin-walled shells is proposed based on the principle of maximum entropy and the Karhunen-Loeve expansion method of random fields. Firstly, the maximum entropy approach is used to estimate the probability density function of the Karhunen-Loeve random variables, which is aimed to model the gemometric imperfections as random fields without the assumpation of Gaussian and homogeneity. Secondly, the quantification of the epistemic uncertainty caused by the availability of only a small size of data on geometric imperfections of thin-walled shells is achieved by extending the classical equationally constrained maximum entropy model to an interval constrained maximum entropy model. Finally, the proposed method is used for imperfection modeling and critical load prediction for A-Shell of the international imperfection databank. It is shown that the proposed random field modeling approach based on the interval constrained maximum entropy principle not only has the ability to quantify the epistemic uncertainty due to small size of data, but also effectively characterizes the higher order moment information of the measured data, Furthermore, it is shown that the Gaussian random field model and the random field model based on the equation constrained maximum entropy principle are the two special cases of the proposed modeling approach in this paper.
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图 2 区间约束极大熵分布与等式约束极大熵分布的熵值比较
Figure 2. Comparison of entropy values between interval constrained maximum entropy distribution and equation constrained maximum entropy distribution
图 3 不同样本量条件下区间约束极大熵分布与等式约束极大熵分布的比较
Figure 3. Comparison of the interval constrained maximum entropy distribution and equation constrained maximum entropy distribution estimated from sample data with different size
图 4 缺陷轴压圆柱薄壳结构随机屈曲分析流程图
Figure 4. Flow chart of stochastic buckling analysis for axial compressive thin-walled cylindrical shell with geometrical imperfection
图 8 Karhunen-Loeve随机变量ξ(i)(i = 1,2,3,4,5,6)的概率密度函数估计结果
Figure 8. PDF estimation of Karhunen-Loeve random variables ξ(i)(i = 1,2,3,4,5,6)
图 9 轴压薄壁圆柱筒的轴压力与轴压位移曲线
Figure 9. Reaction force-displacement curve of axial compressive thin-walled cylindrical shell
表 1 模型参数
Table 1. Model parameters
Radius R/mm Height H/mm Thickness t/mm Young's modulus E/MPa Poisson's ratio ν Density ρ/(mg·mm−3) 101.60 202.29 0.1160 104410 0.3 2 
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表 2 含缺陷薄壁圆柱壳轴压极限载荷无量纲因子αc的统计均值和变异系数
Table 2. Mean and coefficient of variation of the cylindrical shell limit load
Random field model Number of samples Mean Coefficient of variation Gaussian random field 100 0.9041 0.0169 equation constrained maximum
entropy random field100 0.9017 0.0145 interval constrained maximum
entropy random field100 0.9017 0.0153 test sample 7 0.8973 0.0175
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表 3 结构临界载荷KDF估计值
Table 3. KDF of the cylindrical shell limit load
Reliability 0.999 0.99 0.95 Gaussian random field 0.8569 0.8647 0.8803 interval constrained maximum
entropy random field0.8647 0.8698 0.8835 equation constrained maximum
entropy random field0.8707 0.8756 0.8872
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